A Note on the Representation of Bounded Utility Functions Defined on [a, ∞)

The structure and analytical representation of investors' utility-of-wealth functions has long been of interest in portfolio theory. In proposing convenient analytical utility functions most economists have used (i) constant elasticity (power) functions, (ii) the negative exponential function. Both (i) and (ii), of course, restrict the preference structure; Moreover, one may object to (i) because such functions are not uniformly bounded on [0, ∞). And, as has been shown by Arrow [1], this is undesirable in an axiomatic system. The negative exponential function has no such disadvantage, but objections may be raised on empirical grounds. Thus, no simple convenient specification of bounded utility functions on [a, ∞) is available. In fact, even polynomials in wealth of arbitrary order are restrictive since they immediately impose the requirement that moments of wealth are finite. (If the polynomial is of order n, then the nth moment must be finite.)